CS3414 Afterclass Notes --- 29 May, 2002

Fitting Data (parts of Chapters 4, 7, 10)

Introduction (last time)

Polynomial interpolation

Facts (last time)

Choosing basis functions

Usually a bad idea: power basis

Use bi(x) = xi-1

Not usually a good idea because: the interpolant
g(x) can be hard to
evaluate accurately (stability problems, overflow); and
constructin g(x) (i.e., finding the coefficients) is
unnecessarily expensive and ill-conditioned.

A slightly better idea: shifted power basis. The problem of
evaluating g(x) accurately can be improved by using as basis
functions bi(x) = (x-c)i-1, where c is a
constant of roughly the same size as you expect x to be.
This doesn't help improve the problems with constructing g(x) in
the first place, however

A slick idea: Lagrange basis functions

By choosing the basis functions cleverly, we can reduce the
cost of finding constructing g(x) to zero.

See Section 4.1 for the formulas.

While construction of g(x) is trivial when Lagrange basis
functions are used, manipulating g(x) (e.g., evaluating it,
differentiating it, ...) is awkward. And adding new data points
is definitely awkward.

Another slick idea: Newton basis functions

Again, basis functions are chosen so that the cost of
constructing g(x) is relatively cheap; this time
O(n2), which is much better than the
O(n3) which is needed to find the coefficients of
the interpolant with respect to the power basis.